Nonstandard finite difference method for time-fractional singularly perturbed convection–diffusion problems with a delay in time

Abstract

In this work, nonstandard finite difference method is presented for the numerical solution of time-fractional singularly perturbed convection–diffusion problems with a delay in time. The time-fractional derivative is considered in the Caputo sense and discretized using Crank–Nicholson technique. Then, a nonstandard finite difference scheme is constructed on a uniform mesh discretization along the spatial direction. The parameter-uniform convergence of the proposed method is proved rigorously and shown to be $\varepsilon$-uniform convergent with order of convergence $O\left({\left(\Delta t\right)}^2\right)$ along the temporal domain and $M^{-1}$ along the spatial domain. Finally, the proposed scheme is validated using model examples and the computational results are in agreement with the theoretical expectation.